Signals and Systems
Fall 2003
Lecture #8
30 September 2003
1,Derivation of the CT Fourier Transform pair
2,Examples of Fourier Transforms
3,Fourier Transforms of Periodic Signals
4,Properties of the CT Fourier Transform
Fourier’s Derivation of the CT Fourier Transform
x(t) - an aperiodic signal
- view it as the limit of a periodic signal as T → ∞
For a periodic signal,the harmonic components are
spaced ω
0
= 2π/T apart,..
As T → ∞,ω
0
→ 0,and harmonic components are spaced
closer and closer in frequency
Fourier series Fourier integral? →?
Discrete
frequency
points
become
denser in
ω as T
increases
Motivating Example,Square wave
increases
kept fixed
So,on with the derivation,..
For simplicity,assume
x(t) has a finite duration.
Derivation (continued)
Derivation (continued)
a) Finite energy
In this case,there is zero energy in the error
For what kinds of signals can we do this?
(1) It works also even if x(t) is infinite duration,but satisfies:
E.g,It allows us to consider FT for periodic signals
c) By allowing impulses in x(t)or inX(jω),we can represent
even more signals
b) Dirichlet conditions
Example #1
(a)
(b)
Example #2,Exponential function
Even symmetry Odd symmetry
Example #3,A square pulse in the time-domain
Useful facts about CTFT’s
Note the inverse relation between the two widths? Uncertainty principle
Example #4:
x(t) = e
at
2
— A Gaussian,important in
probability,optics,etc.
Also a Gaussian!
Uncertainty Principle! Cannot make
both?t and?ω arbitrarily small.
(Pulse width in t)?(Pulse width in ω)
tω ~ (1/a
1/2
)?(a
1/2
) = 1
CT Fourier Transforms of Periodic Signals
— periodic in t with
frequency ω
o
— All the energy is
concentrated in one
frequency — ω
o
Example #4:
“Line spectrum”
— Sampling functionExample #5:
Same function in
the frequency-domain!
Note,(period in t) T
(period in ω) 2π/T
Inverse relationship again!
Properties of the CT Fourier Transform
1) Linearity
2) Time Shifting
FT magnitude unchanged
Linear change in FT phase
Properties (continued)
3) Conjugate Symmetry
Even
Odd
Even
Odd
The Properties Keep on Coming,..
4) Time-Scaling
a) x(t) real and even
b) x(t) real and odd
c)